11 REGULARIZATION 299 7. Let G be a locally compact group. For each t e R* , let ju,r =£ 0 be a positive bounded measure on G. Suppose that the mapping /K-+JU,, is continuous with respect to the topology $~2 (in the notation of Section 13.20, Problem 1), and that [JLS +1 = p,s * ju,r for all s, t in Rt. (a) Show that there exists a real number e such that ||)Ltt|| = ect, (Observe that the mapping t h-» ||/u,,|| is lower semicontinuous, and apply Problem 6.) (b) Show that, as /-*0, p,t converges (with respect to «^"2) to a Haar measure of total mass 1 on a compact subgroup of G. (Use (12.15.9) to show that there exists a sequence (sn) tending to 0 such that the sequence (fjLSn) tends to a limit p in the top- ology $~2» and that /xr * p, = JJLT = p, * //,, for all t > 0. Deduce that jLts -> JK, as s -»0, and that p * JLC = /x. Complete the proof by using Problem 5 of Section 14.7.) 8. Let A denote Lebesgue measure on R". Let A be a bounded convex open set in R" (Section 8.5, Problem 8), and let D(A) = A — A be the set of all x — y where :c, y e A. The set D(A) is convex, open, symmetric and bounded. For each x ¥= 0 in D(A), let p(x) be the unique real number, belonging to ]0, 1[, such that p(x)"1^ lies in the frontier of D(A). If we put p(0) = 0, then p is a continuous function on D(A) (Section 12.14, Problem 12). (a) For each x e D(A), show that A(A n (A + x)) ^ (I - pW)"A(A). (Observe that if x ^ 0 and p(x)~lx = b — a where a, b E A, then (1 - p(x))A + p(x)b = (1 - p(x))A + p(x)a + x). (b) Show that (A(A))2-f (^A*9A JD<A> (c) Deduce from (a) and (b) that A(A)^f (1-yc JD(A) Prove that this integral is equal to A(D(A)) f1/z/n~10™0w^ = ! (Split up the interval [0, 1] into m parts by means of an increasing sequence (4)o«*«m, and consider D(A) as the union of the sets defined by tk < p(x) < tk+l; then let m -*• oo.) Hence show that 9. Let G be a locally compact group, p a left Haar measure on G, and let V be a sym- metric compact neighborhood of e in G which does not contain any subgroup other than {e}.s a universally measurable